Analytical Pseudoconvexity

THE INSTITUTE OF MATHEMATICS ”SIMION
STOILOW” OF THE ROMANIAN ACADEMY
DOCTORAL THESIS
Summary
Analytical Pseudoconvexity
Advisor,
C.S. I dr. Mihnea Colt¸oiu
Ph. D. Student,
George-Ionut¸ Ionit¸a˘
Bucharest, 2014
Introduction
General notions and fundamental results in the theory of Stein, q-convex
(with corners) and q-complete (with corners) spaces are included in the first
half of the thesis. Also, the specific concepts are presented: convexity with
respect to a linear set introduced by M. Peternell [28], the Andreotti function
of an analytic subset and the definition of a (strictly) q-plurisubharmonic
function (see [15]).
The second part contains the original results (Theorem 2, Theorem 5 and
Theorem 6). This theorems are included in the papers [18], [19] and [20],
which have been accepted for publication.
Theorem 2 generalizes results obtained by Stein [31], Ballico [2], [3], Le
Barz [21] and Vˆajˆaitu [34] and says the following: if π : Z → X is a locally
semi-proper morphism of complex spaces such that X is q-complete, then Z
is (q + r)-complete, where r is the dimension of the fiber.
In the next part there are presented generalizations of the solution to the
Levi problem on complex spaces with isolated singularities given by Colt¸oiu
and Diederich [8]. They proved that if p : Y → X is a Riemann domain,
where X and Y are complex spaces with isolated singularities such that X
is Stein and p is a Stein morphism, then Y is Stein. This result is improved
into two ways:
- we suppose that X is q-complete and we get that Y is q-complete (see
Theorem 5);
- we suppose that the morphism p is locally q-complete with corners and
we get that Y is q-complete with corners (see Theorem 6).
The structure of the thesis. This thesis consists of an introduction,
four chapters and a bibliography with 63 titles.
Acknowledgments
First of all I would like to thank Professor Mihnea Colt¸oiu, my thesis
supervisor, for suggesting me the problems included in the thesis and also
for his patience and his involvement during the last years.
I am very grateful to Professor Cezar Joit¸a who coordinated my Bachelor
and Master thesis at SNS-B. I thank him for the kindness with which he
answered all the questions and for being the one who gave me the opportunity
to start working in Several Complex Variables.
1
I would like to thank my colleague, Ovidiu Preda, with whom I had very
interesting discussions about Stein spaces, q-convexity and locally semi-finite
morphisms.
I am very grateful to the referees: Prof. Dr. George Marinescu (K¨oln
University, Germany), C.S. II Dr. Cezar Joit¸a (”Simion Stoilow” Institute of
Mathematics of the Romanian Academy) and Conf. Dr. Constantin Costara
(”Ovidius” University, Constant¸a, Romania) for their time and effort in reading my thesis manuscript and for their constructive remarks.
Also, I would like to give special thanks to the ”Simion Stoilow” Institute
of Mathematics of the Romanian Academy and to SNS-B. Without their help
I would have not arrived at this point.
Last but not least, I would like to thank my friends and family for their
moral support and their permanent encouragements.
Contents
The central notions in the first chapter are those of a Stein space, (strictly)
plurisubharmonic and q-convex function on an open subset of Cn and those
of a q-convex and a q-complete complex space. An equivalent characterization for a (stricly) plurisubharmonic function is given using the Levi form of
that function. Using local embeddings the concepts of a (strictly) plurisubharmonic and a q-convex function are generalized in the context of complex
spaces.
The following result represents the most important characterization of
Stein spaces.
Theorem 1 ([16] and [23], [24]). A complex space X is Stein if and only if
there exists ϕ : X → R a continuous strongly plurisubharmonic exhaustion
function on X.
The notions of a strictly plurisubharmonic function and of a Stein space
are generalized in the following way.
Definition 1. Let X be a complex space and ϕ ∈ C ∞ (X, R) a smooth function
on X. The function ϕ is called q-convex˘a if for every x ∈ X there exists a
e ⊂ Cn and a function ϕ
e , R) such that ϕ◦ι
local chart ι : U ,→ U
e ∈ C ∞ (U
e = ϕ|U
and with the property that the Levi form of ϕ
e has at least n − q + 1 positive
e.
eigenvalues (> 0) in every point of U
Definition 2. A complex space X is said to be q-convex, if there exists a
compact subset K of X and a smooth exhaustion function ϕ : X → R,
2
which is q-convex on X\K. If we can choose K = ∅, then X is said to be
q-complete.
In the last part of this chapter is presented the concept of a q-convex with
corners function; this function is locally the maximum of a finite number of
q-convex functions (see [12] and [13]).
The second chapter contains the specific results needed in the thesis.
To overcome the problem that neither the sum nor the maximum of two
q-convex functions is not q-convex (as they might have different directions of
positivity), M. Peternell [28] introduced the concept of convexity with respect
to a linear set.
In what follows the notion of the Andreotti function of an analytical
subset A ⊂ X is presented, where X is a complex space. The Andreotti
function will help us get some positive eigenvalues in the ”normal direction”
at the regular points of A.
This chapter ends with a brief presentation of (strictly) q-plurisubharmonic functions.
Chapter 3 contains the first original result of this thesis.
We begin with the remark that if π : Z → X is finite morphism of
complex spaces, then Z is Stein if and only if X is Stein.
In [31], Stein proved that if X and Z are complex spaces and π : Z → X
is an unramified covering such that X is Stein, then Z is Stein.
The last result was generalized to ramified coverings by Le Barz [21]. He
proved that if π : Z → X is a locally semi-finite morphism such that X is
Stein, then Z is Stein.
Ballico [3] improved Stein’s theorem in a different direction: consider
π : Z → X be an unramified covering of complex spaces. If X is q-complete,
then Z is q-complete. Also, in [2], Ballico showed that if π : Z → X is a
finite morphism of complex spaces such that X is q-complete and q-convex,
respectively, then Z is q-complete and q-convex, respectively.
Colt¸oiu and Vˆajˆaitu [11] proved that if π : E → B is a locally analytic
fibration of complex spaces such that the fiber is a Stein curve and B is qcomplete, then E is q-complete. The case when E is a topological covering
of B was already done in [3].
Vˆajˆaitu [34] generalized Ballico’s results in [2] and showed the following
theorem: let π : Z → X be a proper holomorphic map between finite dimensional complex spaces. If X is q-complete, then Z is (q + r)-complete, where
r is the dimension of the fiber.
The purpose of this chapter is to prove a theorem (Theorem 2) which
contains all the results mentioned before. This theorem is included in [18].
3
Following the ideas of Le Barz [21] we give the next definition.
Definition 3. Let X and Z be two complex spaces. We say that a morphism
π : Z → X is
(a) semi-proper if Z is the disjoint union of some open spaces (W m )m∈N
such that π|W m : W m −→ X is proper;
(b) locally semi-proper if for all x ∈ X, there exists a neighbourhood U 3 x
such that π|π−1 (U ) : π −1 (U ) −→ U is a semi-proper morphism.
Now we are ready to state the main result.
Theorem 2 ([18]). Let X and Z be two complex spaces and π : Z → X a
locally semi-proper morphism and r = max{dim π −1 (x) : x ∈ X}. If X is
q-complete, then Z is (q + r)-complete.
The proof of the above mentioned result uses the following criterion for
the q-completeness of a complex space.
Theorem 3 ([10]). Let X be a complex space and M a linear set over X.
Let
[ {Xi }i∈N be an increasing sequence of open subsets of X such that X =
Xi and there are functions ui : Xi → R, ui ∈ B(Xi , M|Xi ) and constants
i∈N
Ci , Di ∈ R, Ci < Di , i ∈ N with the following properties:
(a) {x ∈ Xi : ui (x) < Di } ⊂⊂ Xi for every i ∈ N
(b) {x ∈ Xi+1 : ui+1 (x) < Ci } ⊂ {x ∈ Xi : ui (x) < Di } for every i ∈ N;
(c) for every compact set K ⊂ X there is j = j(K) ∈ N such that
K ⊂ {x ∈ Xi+1 : ui+1 (x) < Ci } for every i ≥ j.
Then there exists an exhaustion function v ∈ B(X, M). In particular, if
codim M ≤ q − 1, then X is q-complete.
The next results contribute essentially in the proof of Theorem 2.
Lemma 1 ([28]). Suppose that ϕ is a q-convex function on a complex space
X. Then there exists a linear set M over X of codimension ≤ q − 1 such
that ϕ is 1-convex with respect to M.
4
e be a local chart of the complex space X
Lemma 2 ([28]). Let ι : U ,→ U
and ϕ : U → R a smooth function. Then ϕ is 1-convex with respect to some
linear set M if and only if for every compact subset K ⊂ U there exists δ > 0
e , R) such that ϕ
and for each x ∈ K there exist ϕ
e ∈ C ∞ (U
e ◦ ι = ϕ and
L(ϕ,
e ι(x))ι∗ (ξ) ≥ δ kι∗ (ξ)k2
for all ξ ∈ Mx .
Proposition 1 ([34]). Let π : Z → X be a holomorphic map. Then there
exists a decreasing chain of p + 1 analytic subsets Ak of Z, where p ≤ dim Z,
Z = Ap ⊃ Ap−1 ⊃ · · · ⊃ A1 ⊃ A0 = ∅ such that for every k ∈ {1, 2, . . . , p}
we have dim Ak−1 < dim Ak , Sing (Ak ) ⊂ Ak−1 and
π|Ak \Ak−1 : Ak \Ak−1 → X
has locally constant rank.
Lemma 3 ([21]). Let X and Z be two complex spaces and π : Z → X
a locally semi-proper morphism. Then there exists a locally finite covering
{Uj }j of Z and a locally finite covering {Vl }l of X such that the following
conditions hold:
1. for all j, there exists a positive integer mj and a local chart ιj : Uj ,→
ej , where U
ej is an open subset of Cmj ;
U
2. for all l, there exists a positive integer nl and a local chart τl : Vl ,→ Vel ,
where Vel is an open subset of Cnl ;
3. for all j, there exists l(j) such that we have π(Uj ) ⊂ Vl(j) and π|Uj
ej → Vel(j) ;
extends to a holomorphic map π
e:U
Also, there exists a C ∞ function f : Z → R such that:
• {z ∈ Z : f (z) < c1 } ∩ {z ∈ Z : (ϕ ◦ π)(z) < c2 } ⊂⊂ Z, ∀c1 , c2 ∈ R;
• for all j, there exists a map gj : Vl(j) → R such that f |Uj = gj ◦ π|Uj ;
• gj has a C ∞ extension, gej : Vel(j) → R;
• for all compact sets K ⊂ X,


2
∂
g
e
j
sup (l(j)) (l(j)) j∈N  ∂zr
∂ z¯s
|
: Vl(j) ∩ K 6= ∅, r, s = 1, nl(j)
τl(j) (Vl(j) ∩K)
5



< ∞.
Lemma 4 ([34]). Let π : Z → X be a holomorphic map between reduced
complex spaces with r = max{dim π −1 (x) : x ∈ X}. Then there exists N
a linear set of codimension ≤ r over Z such that for any relatively compact
open subset U of Z, there exists a finite covering {Vl }l of π(U ) by relatively
compact open subsets and smooth functions ψl : Ul → R+ such that ψl is
1-convex with respect to N over Ul ∩ U , where Ul = π −1 (Vl ).
The chapter ends with a converse theorem of that of Le Barz, namely if X
is n-dimensional Stein space, then there exists f : X → Cn locally semi-finite
morphism. The idea that emerges from the proof of the above mentioned
result is that if f : X → Cn is a locally semi-finite morphism, where X is a
n-dimensional Stein space, then f is almost proper. The converse of the last
statement is studied and it is showed that if f : X → Y is a locally semi-finite
morphism, with Y being Stein, then f is not necessarily almost proper. The
example that is considered is the covering with an infinite number of sheets
of the punctured disk in C, covering given by the logarithm function.
In the last chapter are included the original results from the papers [19]
and [20].
By the solution of the Levi problem if Y ⊂ Cn is an open subset which
is locally Stein, i.e., every point x ∈ Cn has a neighbourhood V such that
V ∩ Y is Stein, then Y is itself Stein (Oka [26] and [27], Bremermann [4],
Norguet [25]). From Oka’s characterization of Stein domains in Cn we have
that in this case − log d is plurisubharmonic on Y , where d represents the
euclidean distance to the boundary of Y . More generally, K. Oka considered
unbranched Riemann domains p : Y → Cn over Cn and he showed that Y is
Stein if and only if − log d is plurisubharmonic on Y . This implies that if p
is a Stein morphism (i.e. each point x ∈ Cn has a neighbourhood V = V (x)
such that p−1 (V ) is Stein), then Y is Stein.
Grauert and Docquier [14] generalized Oka’s result in the context of Stein
manifolds. In particular, they proved that if p : Y → X is an unbranched
Riemann domain, p is a Stein morphism and X is Stein, then Y is Stein.
In [1], Andreotti and Narasimhan showed that if Y ⊂ X is an open
subset of a Stein space X with isolated singularities and Y is locally Stein,
then Y is (globally) Stein. The general case of the above result, for arbitrary
singularities, is called the ”local Steinness problem” or the ”Levi problem on
singular spaces” and it is still an open problem. For a survey concerning the
Levi problem on Stein spaces see [7] and [30].
Colt¸oiu and Diederich improved the above mentioned results and they
proved the following theorem.
Theorem 4 ([8]). Let X and Y be two complex spaces with isolated singu6
larities and p : Y → X an unramified Riemann domain. Suppose that X is
Stein and that p is a Stein morphism, i.e., every point x ∈ X has an open
neighbourhood V = V (x) such that p−1 (V ) is Stein. Then Y is Stein.
The goal of this chapter is to improve Colt¸oiu and Diederich’s theorem
in two directions:
(1) modifying the hypothesis about X being Stein (X will now be qcomplete);
(2) modifying the hypothesis about p : Y → X being a Stein morphism (p
will now be locally q-complete with corners).
The results detailed below are obtained from the above remarks and will
be proved in the subsequent chapters. The theorems have been included in
the papers [19] (Theorem 5) and [20] (Theorem 6), that have been accepted
for publication.
Theorem 5 ([19]). Let X and Y be complex spaces with isolated singularities
and p : Y → X an unbranched Riemann domain. Assume that X is a qcomplete space and that p is a Stein morphism, i.e., each point x ∈ X has
a neighbourhood V = V (x) such that p−1 (V ) is Stein. Then Y is also qcomplete.
Theorem 6 ([20]). Let X and Y be complex spaces with isolated singularities
and p : Y → X an unbranched Riemann domain. Assume that X is Stein
and that p is locally q-complete with corners, i.e., each point x ∈ X has a
neighbourhood V = V (x) such that p−1 (V ) is q-complete with corners. Then
Y is q-complete with corners.
If X and Y are smooth, then Theorem 5 was showed by Vˆajˆaitu in [33].
If p is the inclusion map, then Theorem 6 was proved by Vˆajˆaitu in [32].
Also, if X and Y are smooth, Vˆajˆaitu showed that if X is r-complete with
corners and p is locally q-complete with corners, then Y is (q+r−1)-complete
with corners (see [33]).
Two reductions are made in both proofs: Sing(X) = {x1 , x2 , . . . , xk } is a
finite set and p(Y ) ⊂⊂ X. Also, we distinguish between two cases:
(a) x1 , x2 , . . . , xk ∈
/ p(Y );
(b) x1 , . . . , xl ∈
/ p(Y ) and xl+1 , . . . , xk ∈ p(Y ).
As in the case of Theorem 2, we need a couple of criteria for testing the
q-completeness and the q-completeness with corners of a complex space. We
denote by Fq (X) the set of all q-convex functions with corners on X.
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Proposition 2 ([33]). Let Y be a complex space and N a linear set over Y
of codimension ≤ q − 1. Suppose that there exists a function Φ : Y → R such
that Φ ∈ B(Y, N ) and for every c ∈ R the sublevel set {Φ < c} is N -complete
(i.e., there is an exhaustion function uc ∈ B({Φ < c}, N |{Φ<c} )). Then Y is
q-complete.
Proposition 3 ([33]). Let X be a complex space and Φ ∈ Fq (X) such that
for every c ∈ R the set Xc := {Φ < c} is q-complete with corners. Then X
is q-complete with corners.
The fundamental results that are used in the proofs are Lemma 1 (for
Theorem 5) and the following theorem.
Theorem 7 ([10]). Let X be a 1-convex complex space. Then X carries a
strongly plurisubharmonic exhaustion function Φ : X → [−∞, ∞). Moreover, Φ can be chosen −∞ exactly on the exceptional set S of X and real
analytic outside S.
The ideas that are used in the subsequent are inspired by [8]. We recall here a classical patching technique for plurisubharmonic functions with
bounded differences (using the results of M. Peternell [29] and Matsumoto [22]) and H¨ormander’s [17] regularization method needed to avoid the
difficulties that appear when trying to build a function with a Levi form
bounded from below in the vertical direction. This ideas, as well as Theorem 7, are present in the proofs of both theorems.
For the proof of Theorem 6 we need specific results.
Theorem 8 ([5]). Let X be a complex manifold and f : X → R a continuous
strongly q-plurisubharmonic function. Then for an arbitrary continuous function δ : X → (0, ∞) there exists a function fe ∈ Fq (X) such that |fe − f | < δ.
Lemma 5 ([6]). Consider X to be a complex space, A ⊂ X an analytic
subset and f ∈ Fq (A). Then for every η ∈ C 0 (A, R), η > 0 there
is an open
neighbourhood U of A in X and fe ∈ Fq (U ) such that fe|A − f ≤ η.
Lemma 6 ([28]). Let X be a complex space and A ⊂ X an analytic subset.
Then there exists h ∈ C ∞ (X, R), h ≥ 0 such that:
(a) {h = 0} = A;
(b) for every x ∈ X there exists an open neighbourhood U of x and a smooth
function σ : U → R such that log(h|U \A ) + σ|U \A is plurisubharmonic.
8
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¨
¨
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with parameters, Math. Z. 237 (2001), 815-831.
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(2000), 1-13.
9
[12] Diederich, K., Fornaess, J. E.: Smoothing q-convex functions and vanishing theorems, Invent. Math. 82 (1985), 291-305.
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[26] Oka, K.: Domaines pseudoconvexes, Tohoku Math. J. 49 (1942), 15-52.
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[28] Peternell, M.: Algebraische Variet¨aten und q-vollst¨andige komplexe
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¨
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